Characteristics of pressure gradient force errors in a terrain-following coordinate

A terrain-following coordinate (σ-coordinate) in which the computational form of pressure gradient force (PGF) is two-term (the so-called classic method) has significant PGF errors near steep terrain. Using the covariant equations of the σ-coordinate to create a one-term PGF (the covariant method) can reduce the PGF errors. This study investigates the factors inducing the PGF errors of these two methods, through geometric analysis and idealized experiments. The geometric analysis first demonstrates that the terrain slope and the vertical pressure gradient can induce the PGF errors of the classic method, and then generalize the effect of the terrain slope to the effect of the slope of each vertical layer (φ). More importantly, a new factor, the direction of PGF (α), is proposed by the geometric analysis, and the effects of φ and α are quantified by tan φ·tan α. When tan φ·tan α is greater than 1/9 or smaller than −10/9, the two terms of PGF of the classic method are of the same order but opposite in sign, and then the PGF errors of the classic method are large. Finally, the effects of three factors on inducing the PGF errors of the classic method are validated by a series of idealized experiments using various terrain types and pressure fields. The experimental results also demonstrate that the PGF errors of the covariant method are affected little by the three factors.